We establish two surprising types of Weyl's laws for some compact
RCD(K,N)/Ricci limit spaces. The first type could have power growth
of any order (bigger than one). The other one has an order corrected by
logarithm similar to some fractals even though the space is 2-dimensional.
Moreover the limits in both types can be written in terms of the singular sets
of null capacities, instead of the regular sets. These are the first examples
with such features for RCD(K,N) spaces. Our results depends
crucially on analyzing and developing important properties of the examples
constructed by the last two authors, showing them isometric to the
α-Grushin halfplanes. Of independent interest, this also allows us to
provide counterexamples to conjectures by Cheeger-Colding and by
Kapovitch-Kell-Ketterer.Comment: Final version. To appear in Trans. AMS Series B. 41 page